MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resspsradd Structured version   Unicode version

Theorem resspsradd 17488
Description: A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
resspsr.s  |-  S  =  ( I mPwSer  R )
resspsr.h  |-  H  =  ( Rs  T )
resspsr.u  |-  U  =  ( I mPwSer  H )
resspsr.b  |-  B  =  ( Base `  U
)
resspsr.p  |-  P  =  ( Ss  B )
resspsr.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
Assertion
Ref Expression
resspsradd  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )

Proof of Theorem resspsradd
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 resspsr.u . . 3  |-  U  =  ( I mPwSer  H )
2 resspsr.b . . 3  |-  B  =  ( Base `  U
)
3 eqid 2443 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
4 eqid 2443 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
5 simprl 755 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
6 simprr 756 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
71, 2, 3, 4, 5, 6psradd 17453 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X  oF ( +g  `  H
) Y ) )
8 resspsr.s . . . 4  |-  S  =  ( I mPwSer  R )
9 eqid 2443 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2443 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
11 eqid 2443 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
12 fvex 5701 . . . . . . . 8  |-  ( Base `  R )  e.  _V
13 resspsr.2 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubRing `  R
) )
14 resspsr.h . . . . . . . . . . 11  |-  H  =  ( Rs  T )
1514subrgbas 16874 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
1613, 15syl 16 . . . . . . . . 9  |-  ( ph  ->  T  =  ( Base `  H ) )
17 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1817subrgss 16866 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
1913, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  T  C_  ( Base `  R ) )
2016, 19eqsstr3d 3391 . . . . . . . 8  |-  ( ph  ->  ( Base `  H
)  C_  ( Base `  R ) )
21 mapss 7255 . . . . . . . 8  |-  ( ( ( Base `  R
)  e.  _V  /\  ( Base `  H )  C_  ( Base `  R
) )  ->  (
( Base `  H )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } )  C_  (
( Base `  R )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } ) )
2212, 20, 21sylancr 663 . . . . . . 7  |-  ( ph  ->  ( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
24 eqid 2443 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
25 eqid 2443 . . . . . . 7  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
26 reldmpsr 17428 . . . . . . . . . 10  |-  Rel  dom mPwSer
2726, 1, 2elbasov 14222 . . . . . . . . 9  |-  ( X  e.  B  ->  (
I  e.  _V  /\  H  e.  _V )
)
2827ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( I  e.  _V  /\  H  e.  _V )
)
2928simpld 459 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  I  e.  _V )
301, 24, 25, 2, 29psrbas 17448 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( ( Base `  H )  ^m  { f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
318, 17, 25, 9, 29psrbas 17448 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( Base `  S )  =  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
3223, 30, 313sstr4d 3399 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
3332, 5sseldd 3357 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  ( Base `  S ) )
3432, 6sseldd 3357 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
358, 9, 10, 11, 33, 34psradd 17453 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  oF ( +g  `  R
) Y ) )
3614, 10ressplusg 14280 . . . . . . 7  |-  ( T  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  H ) )
3713, 36syl 16 . . . . . 6  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  H ) )
3837adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  R )  =  ( +g  `  H
) )
39 ofeq 6322 . . . . 5  |-  ( ( +g  `  R )  =  ( +g  `  H
)  ->  oF
( +g  `  R )  =  oF ( +g  `  H ) )
4038, 39syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  oF ( +g  `  R )  =  oF ( +g  `  H
) )
4140oveqd 6108 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  oF ( +g  `  R
) Y )  =  ( X  oF ( +g  `  H
) Y ) )
4235, 41eqtrd 2475 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  oF ( +g  `  H
) Y ) )
43 fvex 5701 . . . . 5  |-  ( Base `  U )  e.  _V
442, 43eqeltri 2513 . . . 4  |-  B  e. 
_V
45 resspsr.p . . . . 5  |-  P  =  ( Ss  B )
4645, 11ressplusg 14280 . . . 4  |-  ( B  e.  _V  ->  ( +g  `  S )  =  ( +g  `  P
) )
4744, 46mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  S )  =  ( +g  `  P
) )
4847oveqd 6108 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X ( +g  `  P ) Y ) )
497, 42, 483eqtr2d 2481 1  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   `'ccnv 4839   "cima 4843   ` cfv 5418  (class class class)co 6091    oFcof 6318    ^m cmap 7214   Fincfn 7310   NNcn 10322   NN0cn0 10579   Basecbs 14174   ↾s cress 14175   +g cplusg 14238  SubRingcsubrg 16861   mPwSer cmps 17418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-tset 14257  df-subg 15678  df-rng 16647  df-subrg 16863  df-psr 17423
This theorem is referenced by:  subrgpsr  17491  ressmpladd  17536
  Copyright terms: Public domain W3C validator